M. Denecker and D. De Schreye. Justification semantics: a unifying framework for the semantics of logic programs. In Proc. of the Logic Programming and Nonmonotonic Reasoning Workshop, pages 365--379, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....16 327996. Annalisa Bossi Dip. di Matimatica Appl. e Informatica Universit a Ca Foscari di Venezia Via Torino, 155 30173 Venezia Mestre, Italy E mail: bossi dsi.unive.it, Phone: 39 41 2908421, Fax: 39 41 2908419. April 1998 Abstract The representation of knowledge in the logic OLP FOL [8], 7] is split in two parts: writing definitions for known concepts, and writing constraints, expressing partial knowledge on other concepts. This is reflected in an OLP FOL theory T , which is a pair: T = T d ; T c ) The definition part T d contains the definitions for known predicates in the ....

....described as a set of tuples of ground atoms with truth value (e.g. fp f ; q u ; r t g, meaning that H I (p) f; H I (q) u and H I (r) t) The truth function of a 2 valued Herbrand interpretation will also be described as a subset of the Herbrand base. We will work in the logic OLP FOL [8], 7] which is an expressive logic to represent uncertainty on definitions of concepts and on the problem domain. A theory T in this logic is a pair (T d ; T c ) of a First Order Logic (FOL) theory T c , the FOL part of T , and a normal open logic program (OLP) T d , the definition part of the ....

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M. Denecker and D. De Schreye. Justification semantics: a unifying framework for the semantics of logic programs. In Proc. of the Logic Programming and Nonmonotonic Reasoning Workshop, pages 365--379, 1993.

....Celestijnenlaan 200A, B 3001 Heverlee, Belgium. sofie cs.kuleuven.ac.be Annalisa Bossi Dip. di Matimatica Appl. e Informatica, Universit a Ca Foscari di Venezia, Via Torino, 155 30173 Venezia Mestre, Italy. bossi dsi.unive.it Abstract The representation of knowledge in the logic OLP FOL [8], 7] is split in two parts: writing definitions for known concepts, and writing constraints, expressing partial knowledge on other concepts. This is reflected in an OLP FOL theory T , which is a pair: T = T d ; T c ) The definition part T d contains the definitions for known predicates in the ....

.... of ground atoms with truth value (e.g. fp f ; q u ; r t g, meaning that H I (p) f; H I (q) u and H I (r) t) The truth function of a 2 valued Herbrand interpretation will also be described as a subset of the Herbrand base (denoting the true atoms) We will work in the logic OLP FOL [8], 7] which is an expressive logic to represent uncertainty on definitions of concepts and on the problem domain. A theory T in this logic is a pair (T d ; T c ) of a First Order Logic (FOL) theory T c , the FOL part of T , and a normal open logic program (OLP) T d , the definition part of the ....

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M. Denecker and D. De Schreye. Justification semantics: a unifying framework for the semantics of logic programs. In L. M. Pereira and A. Nerode, editors, Proc. of the Logic Programming and Nonmonotonic Reasoning Workshop, pages 365--379, 1993.

....theory induces an 6 entailment relation. Given a theory T and formula F based on Sigma, T j= F iff for any Sigma interpretation M such that M j= T , M j= F . Using the above concepts, we define the 3 valued completion semantics for abductive logic programs. This semantics was first defined in [14] and extends the 3 valued completion semantics for non abductive logic programs of [34] to abductive programs and the 2 valued completion semantics for abductive logic programs of [8] with 3 valued models. The semantics of an abductive logic program P A based on Sigma under 3 valued completion ....

....of P A (under 3 valued completion semantics) iff M j= FEQ( Sigma) Comp(P A ) Abd2(P A ) We write P A j= Sigma F to denote that FEQ( Sigma) Comp(P A ) Abd2(P A ) j= F . We write j= Sigma F to denote that FEQ( Sigma) j= F . The 3 valued completion semantics was proposed in [14] and [11] Like most semantics for abductive logic programs, 3 valued completion semantics assigns a 2 valued interpretation to abducible predicates. This is also the case in the 2 valued completion semantics [8] the 2 valued generalised stable semantics [32] and the 3 valued extended ....

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M. Denecker and D. De Schreye. Justification semantics: a unifying framework for the semantics of logic programs. In Proc. of the Logic Programming and Nonmonotonic Reasoning Workshop, pages 365--379, 1993.

....by the loop over negation of p. For programs which do not contain such loops, it can be shown that the undefined predicates can have any interpretation. In other semantics such as the 3 valued completion semantics for abductive programs [4] the justification semantics for abductive programs [7] and the generalised well founded semantics for abductive logic programs [28] even for programs with loops over negation, the interpretation of the undefined predicates can be any. Despite these problems with 2 valued completion semantics, we use it here because of its declarative simplicity and ....

....level mapping. 2 Several types of semantics have been defined for open logic programs: 2 valued completion semantics [3] generalised stable semantics [20] the generalised well founded semantics [28] 3 valued completion semantics and 3 valued (direct) partial) justification semantics with FEQ [7]. Due to the fact that D is acyclic and in each clause of PD , the variables of the body occur in the head, all these semantics coincide in the (weak) sense that the set of all ground atoms implied by D under any of the semantics is identical. This extension of results of [1] is proven formally in ....

M. Denecker and D. De Schreye. Justification semantics: a unifying framework for the semantics of logic programs. In Proc. of the Logic Programming and Nonmonotonic Reasoning Workshop, pages 365--379, 1993.

....[1] proves that the resulting program is acyclic. The same holds for D 0 , and in fact for all transformed domain descriptions: Proposition 3.1 The translation D of any domain description D is acyclic. Several types of semantics have been defined for incomplete programs [3] 14] 19] [7]. Due to the fact that D is acyclic and in each clause of PD , the variables of the body occur in the head, all semantics coincide in the (weak) sense that the set of all ground atoms implied by D under any of the semantics is identical. This extension of results of [1] is proven formally in [4] ....

M. Denecker and D. De Schreye. Justification semantics: a unifying framework for the semantics of logic programs. In Proc. of the Logic Programming and Nonmonotonic Reasoning Workshop, pages 365--379, 1993.

....negative literals. Definition 1 A Positive Inductive Definition is a set D of rules p B with head p 2 D and body B a nonempty set of positive or negative literals 13 This operator is analogous to the TP operator for definite logic programs. 14 Also this formalisation has been used in LP in [9]. 15 This extension facilitates the leap to the case of causal theories with negative literals in the next section. 13 of D. Let Defined(D) be the set of atoms which occur in the head of a rule. For every rule p B 2 D, for every negative literal :q 2 B, it should hold that q 62 Defined(D) ....

M. Denecker and D. De Schreye. Justification semantics: a unifying framework for the semantics of logic programs. Technical Report 157, Department of Computer Science, K.U.Leuven, 1992.

....of rules p(x) B for each x and each set B of p atoms such that M j= F [x; B] meaning that F is true for x in M when p is interpreted as the set B. 4 ffl The model can be expressed also as the interpretation in which each atom has a proof tree. Also this formalisation has been used in LP in [7]. Because it is less commonly used in LP, I present it here for a slightly extended version of the formalism of [2] Let be given a domain D including a subset D o D with symbols t; f 2 D o , an interpretation M interpreting D o (M(t) t; M(f) f) and a definition D, a set of rules p B ....

M. Denecker and D. De Schreye. Justification semantics: a unifying framework for the semantics of logic programs. Technical Report 157, Department of Computer Science, K.U.Leuven, 1992.

....they represent concepts for which no definitions are given. Partial knowledge about these predicates can be expressed in the set of FOL axioms T c . The model semantics of OLP FOL is an extension of the well founded semantics [21] and of the extended well founded semantics [19] and was defined in [11]. This logic has a possible state semantics, that is, a model correspond to a state in which the problem domain might occur according to the (incomplete) expert knowledge (and not a belief set, a set of believed atoms, as in answer set semantics of Extended Logic Programming) At the level of the ....

.... a pair of OLP FOL theories T 1 , T 2 representing the modules of the experts with non intersecting sets of defined predicates, we investigate conditions on T 1 ; T 2 such that: Mod J (T 1 [ T 2 ) Mod J (T 1 ) Mod J (T 2 ) After section 2, which recalls the semantics of OLP FOL from [11], section 3 gives us a first result, stating that for correct theories, the class of models of the composition is contained in the intersection of the classes of models of the two separate theories. In section 4, by using the notion of justification, we give a very general condition, the ....

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M. Denecker and D. De Schreye. Justification semantics: a unifying framework for the semantics of logic programs. In Proc. of the Logic Programming and Nonmonotonic Reasoning Workshop, pages 365--379, 1993.

....languages, bringing them closer to the richer OLP formalism. In fact we notice that recently added constructs to DLs, like reflexive transitive closure of roles, map to more expressive subsets of OLP for which stronger LP semantics than the completion (for example the justification semantics of [10] or the generalized stable model semantics of [16] are appropriate. In sections 2 and 3, we describe syntax and semantics of OLP and Description Logics, respectively. In the fourth section we provide a mapping from DL theories into open logic programs and indicate to which sublanguages of OLP ....

.... , I is a model of if and only if s(I) is a model of T ( It is important to observe here that in DLs no closed world assumption is present: there may be unknown objects that are not mentioned in the theory. This is also possible in OLP, since OLP allows for non Herbrand interpretations (see [10]) Thus representing open domains is possible. 4.2 Description Logics as Sublanguages of OLP The set of open logic programs obtained from DL theories using a mapping like the above one is only a small subset of the set of all possible logic programs. In particular, an open logic program ....

M. Denecker and D. De Schreye. Justification semantics: a unifying framework for the semantics of logic programs. In Proc. of the Logic Programming and Nonmonotonic Reasoning Workshop, pages 365--379, 1993.

....interpretation and the interpretation of the abductive predicates should be two valued. An important property of 3 valued completion semantics is that any (abductive) logic program is consistent. This property is an extension of a result obtained in [24] and was proven for the abductive case in [8, 10]. It is formulated in the following theorem. Theorem 2.1 Let P be an (abductive) logic program based on L. Let I be any 2 valued interpretation of L n fp=n j p=n is a defined predicate of L;P g satisfying FEQ. There exists a model M of comp 3 (L; P ) which extends I, i.e. M restricted to L n ....

....literal. An extension of SLDNFA to deal with floundering negation in positive goals would simply skolemise all variables in a selected non ground negative literal. SLDNFA is not only sound wrt the 3 valued completion semantics but also wrt more fine grained semantics for logic programs. In [10], we presented the justification semantics which extends the well founded semantics [42] and the extended well founded semantics for abductive logic programs [35] In contrast to most current LP semantics, justification semantics is based on general interpretations instead of Herbrand ....

[Article contains additional citation context not shown here]

M. Denecker and D. De Schreye. Justification semantics: a unifying framework for the semantics of logic programs. In Proc. of the Logic Programming and Nonmonotonic Reasoning Workshop, pages 365--379, 1993.

....interpretation and the interpretation of the abductive predicates should be two valued. An important property of 3 valued completion semantics is that any (abductive) logic program is consistent. This property is an extension of a result obtained in [24] and was proven for the abductive case in [8, 10]. It is formulated in the following theorem. Theorem 2.1 Let P be an (abductive) logic program based on L. Let I be any 2 valued interpretation of L n fp=n j p=n is a defined predicate of L;P g satisfying FEQ. There exists a model M of comp 3 (L; P ) which extends I, i.e. M restricted to L n ....

M. Denecker and D. De Schreye. Justification semantics: a unifying framework for the semantics of logic programs. Technical Report 157, Department of Computer Science, K.U.Leuven, 1992.

....Negation As Failure) and the representation of uncertainty on other parts of the application. OLP FOL is derived from Abductive Logic Programming [45] Its semantics and its relationships to other areas in AI such as terminological logic and iterated inductive definitions have been studied in [31, 27, 28, 70]. We showed its expressivity for representing uncertainty in [35, 32, 27] With respect to the representation of temporal information, starting from the original event calculus [47] we have developed Open Event Calculus, an OLP FOL theory describing time, action and state. The resulting theory ....

M. Denecker and D. De Schreye. Justification semantics: a unifying framework for the semantics of logic programs. In Proc. of the Logic Programming and Nonmonotonic Reasoning Workshop, pages 365--379, 1993.

....is that it is the weakest semantics known for the (O)LP formalism. In [4] the following theorem is proven: Theorem 3. 2 If M is a model of P D wrt (2 valued completion semantics [2] 3] generalised stable semantics [14] generalised well founded semantics [22] justification semantics [5]) then M is a model of P D wrt 3 valued completion semantics. As a consequence, the 3 valued completion semantics induces the weakest entailment relation j= if an open theory entails F according to the 3 valued completion semantics then also wrt to the other semantics. In intuitive terms: the ....

....and quaker) OLP )FOL does not provide default rules; therefore, we believe that the most precise formalisation in OLP FOL would be the following FOL theory: 10 It is well known that when Sigma contains functors with arity 0, the DCA can not be represented correctly in FOL. As was shown in [5], the DCA can be represented correctly in OLP FOL under stronger semantics than the 3 valued completion semantics. T 1 = f republican(nixon) quaker(nixon) 8X: hawk(X) dove(X) g in which the two defaults are not represented. However, 24] argued that though FOL is unsuited to represent ....

[Article contains additional citation context not shown here]

M. Denecker and D. De Schreye. Justification semantics: a unifying framework for the semantics of logic programs. In Proc. of the Logic Programming and Nonmonotonic Reasoning Workshop, pages 365--379, 1993.

....presentation of [2] inductive definitions are dually defined as monotonic TP like operators. This is the common way in fixpoint logic (hence the name) The model can be expressed also as the interpretation in which each atom has a proof tree. Also this formalisation has been used in LP in [7]. Because it is less commonly used in LP, I present it here for a slightly extended version of the formalism of [2] Let be given a symbol domain D, including a subset D o D which includes the truth values t; f, an interpretation M interpreting the symbols of D o such that M (t) t; M (f) f. ....

M. Denecker and D. De Schreye. Justification semantics: a unifying framework for the semantics of logic programs. Technical Report 157, Department of Computer Science, K.U.Leuven, 1992.

....vice versa. The example shows that the meaning of an abductive program is only defined when the undefined predicates are specified. It shows also that logic programs are special abductive logic programs providing definitions for all predicates. We have investigated this view in earlier work. In [6], we proposed a framework to investigate the extent to which the notion of definition is supported by the existing semantics of (abductive) logic programs. In [7] we have conducted an experiment in using the abductive logic programming formalism for representing uncertainty in the context of ....

....the lack of a declarative semantics for the formalism. In the past many model semantics have been defined for abductive logic programming, for example the generalised stable semantics [13] the completion semantics for abductive logic programs [2] and the extended well founded semantics [22] In [6], we proposed the justification semantics. It is important to realise that all except the justification semantics were introduced from the nonmonotonic perspective on logic programming. These semantics aimed to be abstract characterisations of what abductive solutions should be computed by an ....

M. Denecker and D. De Schreye. Justification semantics: a unifying framework for the semantics of logic programs. In Proc. of the Logic Programming and Nonmonotonic Reasoning Workshop, pages 365--379, 1993.

....Indeed, a Herbrand interpretation of Sigma i is a special case of a general interpretation of Sigma i . The restriction of a Herbrand interpretation of Sigma i to the alphabet Sigma defines a general, non Herbrand interpretation of Sigma. This compromise is only partially successful. In [4], an example is given showing that the use of general interpretations is strictly more expressive than the solution in [10] 8 Conclusion The open logic program formalism is an expressive declarative logic in its own right. Theory and examples in this paper show the role of logic programs as sets ....

M. Denecker and D. De Schreye. Justification semantics: a unifying framework for the semantics of logic programs. Technical Report 157, Department of Computer Science, K.U.Leuven, 1992.

....by the loop over negation of p. For programs which do not contain such loops, it can be shown that the abductive predicates can have any interpretation. In other semantics such as the 3 valued completion semantics for abductive programs [4] the justification semantics for abductive programs [7] and the generalised well founded semantics for abductive logic programs [27] even for programs with loops over negation, the interpretation of the undefined predicates can be any. Despite these problems with 2 valued completion semantics, we use it here because of its declarative simplicity and ....

....is a level mapping. 2 Several types of semantics have been defined for open programs: 2 valued completion semantics [3] generalised stable semantics [20] the generalised well founded semantics [27] 3 valued completion semantics and 3 valued (direct) partial) justification semantics with FEQ[7]. Due to the fact that D is acyclic and in each clause of PD , the variables of the body occur in the head, all these semantics coincide in the (weak) sense that the set of all ground atoms implied by D under any of the semantics is identical. This extension of results of [1] is proven formally ....

M. Denecker and D. De Schreye. Justification semantics: a unifying framework for the semantics of logic programs. In Proc. of the Logic Programming and Nonmonotonic Reasoning Workshop, pages 365--379, 1993.

....of (Aczel 1977) inductive definitions are dually defined as monotonic TP like operators. This is the common way in fixpoint logic (hence the name) ffl The model can be expressed also as the interpretation in which each atom has a proof tree. Also this formalisation has been used in LP in (Denecker De Schreye 1992). Because it is less commonly used in 4 Definitions represented in the other style can be represented in this abstract way. Given the mathematical structure M and formula F [X; p] define the domain D as the set of atoms p(x) with x 2 M n . Define D as the set of rules p(x) B for each x ....

Denecker, M., and De Schreye, D. 1992. Justification semantics: a unifying framework for the semantics of logic programs. Technical Report 157, Department of Computer Science, K.U.Leuven.

....the declarative reading of logic programs as sets of definitions and we evaluate the three families of semantics as formalisations of this declarative reading. The paper ends with a discussion (section 9) and a conclusion (section 10) A short paper with the main technical results was published as [7]. 2 Preliminaries A formal logic is a pair of its syntax and its semantics. The syntax or the formalism describes how well formed formulas are constructed. The semantics describes the meaning of the formulas and of sets of formulas. The formalism on which we focus consists of two components: the ....

....the fact and its negation. Justification semantics and well founded semantics use an elegant solution for dealing with such badly defined facts: their truth value is defined as u, which can be read as badly defined. The occurrence of u in a model points to a bug in the definition. Therefore, in [7] we proposed to interpret u as locally inconsistent. For the above reasons, we argue that justification semantics supports the declarative reading of logic programs as definitions, even when they contain recursion over negation. If the DCA holds then by theorem 7.5, well founded semantics ....

M. Denecker and D. De Schreye. Justification semantics: a unifying framework for the semantics of logic programs. In Proc. of the Logic Programming and Nonmonotonic Reasoning Workshop, pages 365--379, 1993.

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